三维Navier-Stokes方程随机拉格朗日表示中韦伯场的精确均值-协方差动力学

中文摘要 AI 辅助

康斯坦丁-艾耶尔公式将三维环面$\boldsymbol{\top}^3$上不可压缩Navier-Stokes方程的光滑解表示为$u=\boldsymbol{\top}\boldsymbol{\top}[(\nabla A_t)^\top(u_0\bigcirc A_t)]$，即随机韦伯场的投影期望。我们将该期望分解为均值与中心化协方差的乘积，证明该协方差连同逆变形梯度的二阶矩和两点协方差满足一个封闭的确定性平流-扩散方程：由于每个拉格朗日标记由同一条布朗路径驱动，二次协方差坍缩为拉普拉斯算子，且不会出现矩层级。从这些方程中，我们得到逐点和赫尔德协方差界的能量/生产平衡，以及分离变量中两点算子的精确退化。另一组结果涉及平均位移，其沿流动方向满足受迫热方程；其$L^2$范数由初始能量无条件控制，而一组精确的Navier-Stokes剪切流表明，不存在可一致保持到$t=0$的类似赫尔德界。我们进一步分离出若干障碍，每个障碍都有明确的反例支撑——即使对于光滑无散漂移的真实随机流，期望变形梯度的范数也不控制期望范数；平均的空间赫尔德界不蕴含抛物型坎帕纳托衰减——我们证明了一个与速度的塞尔林范数等价的规范不变商准则，以及关于平均坐标仿射秩至少为1的古老解的刘维尔型刚性定理。我们未对全局正则性作出断言；缺失的步骤被明确表述为开放问题。

英文摘要

The Constantin-Iyer formula represents a smooth solution of the incompressible Navier-Stokes equations on $\mathbb{T}^3$ as $u=\mathbb{P}\,\mathbb{E}[(\nabla A_t)^\top(u_0\circ A_t)]$, the projected expectation of a stochastic Weber field. We separate this expectation into the product of the means and a centred covariance, and show that the covariance -- together with the second moment of the inverse deformation gradient and the two-point covariance -- satisfies a closed deterministic advection-diffusion equation: because every Lagrangian label is driven by the same Brownian path, the quadratic covariation collapses into a Laplacian and no moment hierarchy appears. From these equations we obtain an energy/production balance, pointwise and Holder covariance bounds, and an exact degeneracy of the two-point operator in the separation variable. A second group of results concerns the mean displacement, which solves a forced heat equation along the flow; its $L^2$ norm is controlled unconditionally by the initial energy, while an exact family of Navier-Stokes shear flows shows that no analogous Holder bound can hold uniformly down to $t=0$. We further isolate several obstructions, each backed by an explicit counterexample -- the norm of an expected deformation gradient does not control the expected norm, even for genuine stochastic flows of smooth divergence-free drifts; a spatial Holder bound on the mean does not imply parabolic Campanato decay -- and we prove a gauge-invariant quotient criterion equivalent to the Serrin norm of the velocity, together with a Liouville-type rigidity theorem for ancient solutions whose mean coordinate is affine of rank at least one. No claim is made regarding global regularity; the missing steps are stated explicitly as open problems.

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12 pages